There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

Ever since its discovery, the Berry phase has permeated through all branches of physics. Over the last three decades, it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. We start with a brief summary of necessary background, followed by a detailed discussion of the Berry phase effect in a variety of solid state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, we demonstrate a re-quantization method that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as a basic ingredient to our understanding of basic material properties.

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Berry phase effects on electronic properties

A scheme that reduces the calculations of ground-state properties of systems of interacting particles exactly to the solution of single-particle Hartree-type equations has obvious advantages. It is not surprising, then, that the density functional formalism, which provides a way of doing this, has received much attention in the past two decades. The quality of the energy surfaces calculated using a simple local-density approximation for exchange and correlation exceeds by far the original expectations. In this work, the authors survey the formalism and some of its applications (in particular to atoms and small molecules) and discuss the reasons for the successes and failures of the local-density approximation and some of its modifications.

The density functional formalism, its applications and prospects

Electronic excitations lie at the origin of most of the commonly measured spectra. However, the first-principles computation of excited states requires a larger effort than ground-state calculations, which can be very efficiently carried out within density-functional theory. On the other hand, two theoretical and computational tools have come to prominence for the description of electronic excitations. One of them, many-body perturbation theory, is based on a set of Green’s-function equations, starting with a one-electron propagator and considering the electron-hole Green’s function for the response. Key ingredients are the electron’s self-energy S and the electron-hole interaction. A good approximation for S is obtained with Hedin’s GW approach, using density-functional theory as a zero-order solution. First-principles GW calculations for real systems have been successfully carried out since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeter equation, via a functional derivative of S. An alternative approach to calculating electronic excitations is the time-dependent density-functional theory (TDDFT), which offers the important practical advantage of a dependence on density rather than on multivariable Green’s functions. This approach leads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than a four-point, interaction kernel. At present, the simple adiabatic local-density approximation has given promising results for finite systems, but has significant deficiencies in the description of absorption spectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, and appreciable distortions of the spectral line shapes. The search for improved TDDFT potentials and kernels is hence a subject of increasing interest. It can be addressed within the framework of many-body perturbation theory: in fact, both the Green’s functions and the TDDFT approaches profit from mutual insight. This review compares the theoretical and practical aspects of the two approaches and their specific numerical implementations, and presents an overview of accomplishments and work in progress.

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Electronic excitations: density-functional versus many-body Green's-function approaches

75 APPENDIX E. 77 APPENDIX F. 79 APPENDIX G. 82 APPENDIX H. Introduction. Generalized Green's functions. Equations of motion for the generalized single-particle Green's function. Bethe-Salpeter equation for the two-particle Green's function. Connection with linear-response theory. Conserving approximations. Gauge invariance, current conservation, and the Ward identities. Optical properties of semiconductors. Single-particle energy levels. Screening of static impurities. Excitonic states. A functional derivative identity. Dyson's equation. Reducible vs. irreducible parts of the correlation functions. Elimination of the spin variables and transformation to the energy-momentum representation. The Thomas-Reiche-Kuhn sum rule. The Clausius-Mossotti relation. The particle-hole Green's function. The Haken potential.

http://bcsbec.df.unicam.it/files/LaRNC11_N12(1988).pdf

Application of the Green's functions method to the study of the optical properties of semiconductors

An ultracold gas of strongly interacting fermions exhibits a pseudogap phase in which pairs of fermions exist above the superfluid transition, but lack the phase coherence of a superfluid.

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Observation of pseudogap behaviour in a strongly interacting Fermi gas

A theory of core excitons in semiconductors is formulated, taking into account the frequency dependence of the dielectric matrix which screens the electron-hole attraction. The present approach combines standard many-body techniques (which reduce the Bethe-Salpeter equation for the two-particle Green's function to an effective eigenvalue problem) with elements drawn from Fano's formalism for discrete states interacting with continuum channels. The positions and the widths of core-exciton resonances are affected by dynamical screening, which increases the binding energy above its value for static screening and decreases its Auger width below its value for a core hole. The latter effect is peculiar to a dynamical theory and has recently been confirmed experimentally.

Effects of dynamical screening on resonances at inner-shell thresholds in semiconductors

The BCS–BEC crossover: From ultra-cold Fermi gases to nuclear systems

A central problem in one-electron band calculations is the proper inclusion of exchange and correlations. We have performed a first-principles calculation by utilizing the Green's-function method with an energy-dependent nonlocal self-energy operator obtained by replacing the Coulomb potential in the exchange operator by a dynamically screened interaction. Diamond was chosen as a prototype of covalent materials. To be consistent with a variety of experimental facts, we have taken the dielectric matrix of the medium within the time-dependent screened Hartree-Fock approximation, thereby including both local-field and electron-hole (excitonic) effects. Previous calculations along similar lines have been restricted either to a random-phase-approximation frequency-independent dielectric function or to a plasmon-pole approximation. We have investigated for the first time the role of a realistic frequency and wave-vector-dependent dielectric matrix, and examined the relative importance of the electron-hole excitations and of the plasma resonance across the range of the valence and conduction bands. The correlated band structure was calculated by diagonalizing the quasiparticle equation of motion in a local orbital basis in order to exploit the local character of both the self-energy operator and the orbitals spanning these bands. We have found that the plasma resonance does not contribute appreciably in the energy range about the band gap while it contributes significantly to the valence bandwidth. Our values of 7.4 eV (band gap) and 25.2 eV (valence band-width) are in good agreement with reflectivity and photoemission experiments. Implications for the local-density and the energy-independent Coulomb-hole plus screened-exchange approximations are discussed. In addition, our method, by utilizing an energy-dependent self-energy, has also enabled us to calculate quasiparticle damping times (specifically, intraband Auger decay rates) that are consistent with photoemission spectra.

G. C. Strinati

Papers

Application of the Green's functions method to the study of the optical properties of semiconductors

Observation of pseudogap behaviour in a strongly interacting Fermi gas

Effects of dynamical screening on resonances at inner-shell thresholds in semiconductors

The BCS–BEC crossover: From ultra-cold Fermi gases to nuclear systems

Dynamical aspects of correlation corrections in a covalent crystal